Tuesday, May 5, 2015

Existential commitments of First Order Logic

In First Order Logic (FOL), we have two oddities: (a) if "b" is a name, then it's a theorem that b exists, and that (b) it's a theorem that something or other exists. We might conclude that since theorems hold necessarily, everything that exists, exists necessarily. Or we might be embarrassed and reject FOL, going for some version of free logic.

Maybe, though, what we should say is that just as ordinary language sentences have presuppositions, a language can have presuppositions. Presuppositions make communication easier. Instead of a nurse's making the convoluted request "If you have an age, please tell me your age; otherwise, please tell me that you're ageless", the nurse can simply presuppose that you have an age and ask: "How old are you?" It's not particularly surprising that presuppositions might also make reasoning easier. It can be easier to reason on the presupposition that there is something, and on the presupposition that names have reference. So FOL has presuppositions. No need for embarrassment: the presuppositions make things simpler for us, much as it's easier to work with commutative groups than groups in general.

Of course, if a language L has presuppositions, then we shouldn't expect its theorems to hold necessarily. Rather, a theorem is something that necessarily follows from the presuppositions. We can without embarrassment say that it's a theorem of an appropriate dialect of FOL that Obama exists, since the only modal conclusion we can make is that, necessarily, if the presuppositions of the dialect are true, Obama exists.

We could search for a logic without presuppositions. That's a worthwhile quest, and leads to exploring various free logics. But we shouldn't go overboard in worrying about the metaphysical consequences if we don't find a good one. Likewise, we shouldn't worry too much if we can't find a satisfactory quantified modal logic. These are just tools. Nice to have, but people have done just fine with modal and other arguments for centuries without much of a formal logic.

It follows from the first theorem that there is something. Not so ontologically worrisome: I think it's a necessary truth that there is something. If there are no empty names, then it follows trivially (almost) that all of your names name something, perhaps the same thing. That's a bigger deal. It's useful not to be committed to things you're trying to show don't exist! Certainly, atheists don't want an ontological commitment to God simply in virtue of offering an atheological argument. And counterfactual reasoning won't help here.

In my Berkeley book (currently under review at OUP), I argue that in Berkeley's view sign systems (including languages) can build in substantive assumptions about what the world is like. This raises the possibility that sentences that are 'analytic' in the sense of being guaranteed assertable by linguistic rules might turn out to be false. I suggest as examples 'combustion involves the release of phlogiston' (authorized by the convention for the use of 'phlogiston') or 'Xs are bad people', where 'X' is a racial slur (the fact that this assumption about the group is built into the meaning of 'X' is what makes it a slur). For Berkeley, truth and falsity come in degrees, and all real human languages have incorrect assumptions built into them (to a greater or lesser degree), which means that all of our assertions are to some degree false (some are only a little false while others are VERY false).

Here's one of the main Berkeley texts I use:The work of science and speculation is to unravel our prejudices and mistakes, untwisting the closest connexions, distinguishing things that are different, instead of confused and perplexed, giving us distinct views, gradually correcting our judgment, and reducing it to a philosophical exactness. And, as this is the work of time, and done by degrees, it is extremely difficult, if at all possible, to escape the snares of popular language, and the being betrayed thereby to say things strictly speaking neither true nor consistent . . . For, language being accommodated to the prænotions of men and use of life, it is difficult to express therein the precise truth of things, which is so distant from their use, and so contrary to our prænotions. (TVV 35)

Also see PHK 52 and Siris 296.

So I think Berkeley's 'praenotions' are just what you have in mind as presuppositions of whole languages (rather than just individual sentences).

In the vicinity, there are two other things. First, Prior's tonk connective. Plausibly the validity of the rules of inference associated with a connective is a presupposition of the language.

Second, Tarski's claim that ordinary language is incoherent because it has an unrestricted truth predicate. The literature considers this to be a mysterious claim. But if we take a language to come with presuppositions, then Tarski's claim can be unpacked neatly as: The presuppositions of ordinary language are incoherent.

I think I remember Jim Van Cleve bringing up tonk when I was writing the part about inference in my dissertation (on which this book is based), but I didn't end up discussing that in the final version. I hadn't thought of the Tarski connection. That's very interesting!

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I am a philosopher at Baylor University. This blog, however, does not purport to express in any way the opinions of Baylor University. Amateur science and technology work should not be taken to be approved by Baylor University. Use all information at your own risk.